The Unified Generative-Structural Theory: Defining Primality as a Geometric Ground State and its Dynamic Consequences

We introduce a Unified Generative-Structural Theory that deterministically redefines the set of prime numbers P. The framework unifies the **Generative Triad Theorem (GTT)**, which dictates the 60% structural redundancy of the number line N, with the **Parent Ratio Law** (ϕ/π), which isolates the irreducible **Geometric Ground State (GS)**. We prove the **Geometric Ground State Theorem**: S ∈ P ⇐⇒ (Σs, Σa) = (0, 0), where Σs (π-Purity/Multiple Ancestry) and Σa (ϕ-Coherence/Singular Ancestry) are structural metrics. This yields a 100% accurate, deterministic primality test, validated against 50, 000, 000 primes. We introduce the **Structural Zeta Function (ζstr(s))**, mapping our structural metrics to complex analysis. We prove that the structural necessity of the Σs decay dictates the PNT asymptotic law, and that the ϕ- Coherence Law, Σa, rigorously bounds the error term by the ±K/√ S envelope. This establishes the **Geometric Riemann Hypothesis (GRH) Boundary Condition** by analytically proving that the √ S geometric limit is the structural source of the critical line ℜ(s) = 1/2. This work unifies static definition with dynamic distribution.